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Mirrors > Home > NFE Home > Th. List > proj2op | Unicode version |
Description: The second projection operator applied to an ordered pair yields its second member. Theorem X.2.8 of [Rosser] p. 283. (Contributed by SF, 3-Feb-2015.) |
Ref | Expression |
---|---|
proj2op | Proj2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-op 4566 | . . . . 5 Phi Phi 0c | |
2 | 1 | eleq2i 2417 | . . . 4 Phi 0c Phi 0c Phi Phi 0c |
3 | elun 3220 | . . . . 5 Phi 0c Phi Phi 0c Phi 0c Phi Phi 0c Phi 0c | |
4 | vex 2862 | . . . . . . . . 9 | |
5 | 4 | phiex 4572 | . . . . . . . 8 Phi |
6 | snex 4111 | . . . . . . . 8 0c | |
7 | 5, 6 | unex 4106 | . . . . . . 7 Phi 0c |
8 | eqeq1 2359 | . . . . . . . 8 Phi 0c Phi Phi 0c Phi | |
9 | 8 | rexbidv 2635 | . . . . . . 7 Phi 0c Phi Phi 0c Phi |
10 | 7, 9 | elab 2985 | . . . . . 6 Phi 0c Phi Phi 0c Phi |
11 | phi011 4599 | . . . . . . . . 9 Phi 0c Phi 0c | |
12 | equcom 1680 | . . . . . . . . 9 | |
13 | 11, 12 | bitr3i 242 | . . . . . . . 8 Phi 0c Phi 0c |
14 | 13 | rexbii 2639 | . . . . . . 7 Phi 0c Phi 0c |
15 | eqeq1 2359 | . . . . . . . . 9 Phi 0c Phi 0c Phi 0c Phi 0c | |
16 | 15 | rexbidv 2635 | . . . . . . . 8 Phi 0c Phi 0c Phi 0c Phi 0c |
17 | 7, 16 | elab 2985 | . . . . . . 7 Phi 0c Phi 0c Phi 0c Phi 0c |
18 | risset 2661 | . . . . . . 7 | |
19 | 14, 17, 18 | 3bitr4i 268 | . . . . . 6 Phi 0c Phi 0c |
20 | 10, 19 | orbi12i 507 | . . . . 5 Phi 0c Phi Phi 0c Phi 0c Phi 0c Phi |
21 | 3, 20 | bitri 240 | . . . 4 Phi 0c Phi Phi 0c Phi 0c Phi |
22 | 2, 21 | bitri 240 | . . 3 Phi 0c Phi 0c Phi |
23 | phieq 4570 | . . . . . 6 Phi Phi | |
24 | 23 | uneq1d 3417 | . . . . 5 Phi 0c Phi 0c |
25 | 24 | eleq1d 2419 | . . . 4 Phi 0c Phi 0c |
26 | df-proj2 4568 | . . . 4 Proj2 Phi 0c | |
27 | 4, 25, 26 | elab2 2988 | . . 3 Proj2 Phi 0c |
28 | 0cnelphi 4597 | . . . . . . . 8 0c Phi | |
29 | ssun2 3427 | . . . . . . . . . 10 0c Phi 0c | |
30 | 0cex 4392 | . . . . . . . . . . 11 0c | |
31 | 30 | snid 3760 | . . . . . . . . . 10 0c 0c |
32 | 29, 31 | sselii 3270 | . . . . . . . . 9 0c Phi 0c |
33 | eleq2 2414 | . . . . . . . . 9 Phi 0c Phi 0c Phi 0c 0c Phi | |
34 | 32, 33 | mpbii 202 | . . . . . . . 8 Phi 0c Phi 0c Phi |
35 | 28, 34 | mto 167 | . . . . . . 7 Phi 0c Phi |
36 | 35 | a1i 10 | . . . . . 6 Phi 0c Phi |
37 | 36 | nrex 2716 | . . . . 5 Phi 0c Phi |
38 | 37 | biorfi 396 | . . . 4 Phi 0c Phi |
39 | orcom 376 | . . . 4 Phi 0c Phi Phi 0c Phi | |
40 | 38, 39 | bitri 240 | . . 3 Phi 0c Phi |
41 | 22, 27, 40 | 3bitr4i 268 | . 2 Proj2 |
42 | 41 | eqriv 2350 | 1 Proj2 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wo 357 wceq 1642 wcel 1710 cab 2339 wrex 2615 cun 3207 csn 3737 0cc0c 4374 cop 4561 Phi cphi 4562 Proj2 cproj2 4564 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj2 4568 |
This theorem is referenced by: opth 4602 opexb 4603 |
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